Communication media that are distortive, such as an optical fiber, are often used in high-speed data communication. In some cases, the distortion introduced by the communication medium is nonlinear. In order to achieve a high data rate or a high bandwidth, multi-mode communication is often employed in which different data signals are simultaneously transmitted via different channels, also called modes, of the communication medium. The carrier frequencies used for different modes are typically different.
One important physical phenomenon currently limiting the information capacity of optical fibers over long distances when multi-mode communication is employed has been identified as the nonlinear interactions between different frequency modes, or cross talk across two or more channels. When multiple signals with initial profiles {Al(0, t)}l=1l where l is the total number of modes are sent over multiple channels of a medium, those signals generally interact with each other over long distances due to the presence of weak nonlinearities, and distort the signal(s) in one or more other channels. This produces distorted signals {Al(L, t)}l=1l at the receiver which can bear little resemblance to the original transmitted signals, and this impedes their appropriate decoding at the receiver.
According to one technique commonly employed to reduce the effect of crosstalk, the power of the data signals when they are launched into the communication medium is reduced, so that the effect of crosstalk may be reduced. The low-power or low strength signals generally do not propagate effectively over a long distance, however. Long in this context can be a few hundreds of meters, a few kilometers, tens, hundreds, or thousands of kilometers, etc. Therefore, in order to transmit the signals over long distances several relays (also called repeaters) are generally needed. The relays may be provided every few hundreds of meters or a few kilometers, or tens or hundreds of kilometers, so that any nonlinear effects can minimized between the relay stations, reducing signal distortion at the final receiver. Relay stations can be expensive and generally increase the cost and complexity of the communication system.
Nonetheless, it has been demonstrated recently that the combination of highly-correlated co-polarized frequency modes with a signal pre-compensation step can significantly reduce the need for such expensive and inefficient relay stations. Such pre-compensation involves finding appropriate initial profiles such that the signal reaching the receiver has the desired form despite the presence of nonlinearities in the dispersive communication medium. In general, in the computations performed to achieve pre-compensation, the Nonlinear Schrodinger Equation (NLSE) modeling the communication medium is solved numerically.
For example, one benchmark, the split-symmetric method, is a time-stepping method that can be written as follows:u(z,t+δt)=−1[e−iδtk2[eiδtN(u(z,t))u(z,t)]]where u(z, t) is a time-harmonic solution to the nonlinear Schrödinger equation with wave number k, and  is the Fourier transform. A deeper look at the equation above shows that if NzSS and NtSS points are respectively used in evaluating the z and t variables, representing the propagation distance and propagation time, respectively, and if the fast Fourier transform (FFT) is used, the final cost of the computations is:O(NtSSNzSS log(NzSS))
For typical communication systems, this often leads to large problems involving hundreds of millions of unknowns, and such problems cannot be solved fast enough to provide real-time pre-compensation. We note that some efforts have been made recently towards reducing the complexity of solving the NLSE and computing pre-compensated signals. However, these proposed methods are either based upon heuristics, or they are lower-order methods, where accuracy of the solution can be increased by decreasing the step size, which can disproportionately increase the computation time. Just as in the case of the split-symmetric method, this can significantly increase the number of unknowns required to reach an acceptable level of accuracy. As such, various known techniques do not provide a numerical solution fast enough, e.g., within a few seconds, a few milliseconds, or in even less time, for real-time pre-compensation of signals to be transmitted via a communication medium.